We show that any pseudoentangled state ensemble with a gap of $t$ bits of entropy requires $\mathrm{\Omega }\left(t\right)$ non-Clifford gates to prepare. This bound is tight up to constant factors if linear-time quantum-secure pseudorandom functions exist. Our result follows from a polynomial-time algorithm to estimate the entanglement entropy of a quantum state across any cut of qubits. When run on an $n$-qubit state that is stabilized by at least $2^{n-t}$ Pauli operators, our algorithm produces an estimate that is within an additive factor of $\frac{t}{2}$ bits of the true entanglement entropy.

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• Quantum algorithms
• Quantum algorithms & computation
• Quantum computation
• Quantum cryptography
• Quantum information processing
• Quantum information theory